Thursday, 10 August 2017

ALGEBRA 4


Expand 10w(5w – 3)

solution

= 10w(5w – 3)

= (10w x 5w) – (10w x 3)

= 50w2 – 30w answer

TRY THIS………..


Expand 4y(9y+ 10)  


VECTORS 2


If u = 20i + 2j and v = 3i + 10j find 5u + 3v

solution

= 5u + 3v

= 5(20i + 2j) + 3(3i + 10j)

= 100i + 10j + 9i + 30j

= (100i + 9i) + (10j + 30j)

= 109i + 40j

Hence 5u + 3v = 109i + 40j .

TRY THIS..................

NECTA 2001 QN. 12a (i)


If u = 4i + 3j and v = 2i + 4j;  find 2u + 3v.

FUNCTIONS 2


If F(x) = log2x, Find F(1/ 8192)

Solution

F(x) = log2x

F(1/8192) = log2(1/ 8192)

F(1/8192) = log2 8192-1

F(1/8192) = log2(213)-1

F(1/8192) = log22(13 x -1)

F(1/8192) = log22-13

F(1/8192) = -13log22

F(1/8192) = -13 x 1

F(1/8192) = -13

Therefore F(1/ 8192) = -13


TRY THIS…………………


If F(x) = log5x, Find F(1/3125)

PROBABILITY 1


A number is chosen at random from 1 – 15 inclusive. Find the probability that it is a multiple of five or an even number greater than 8.

Solution

THIS IS A NON-MUTUALLY EXCLUSIVE EVENT.

Let n(S) represent sample space
P(E) = probability of even number greater than 8
P(M) = probability of a multiple of 5

n(S) = {1, 2, 3, 4, 5, 6, 7, 8, 9,10, 11, 12, 13, 14, 15} = 15
n(E) = {10, 12, 14} = 3
n(M) = {5, 10, 15} = 3
But 10 appears on both categories.
P(E) = 3/15
P(M) = 3/15
P(EnM) = 1/15
………………………….
P(EuM) = P(E) + P(M) - P(EnM)  

P(EuM) = 3/15 + 3/15 -1/15

              = 5/15

              = 1/3  after simplification

P(EuM) = 1/5

TRY THIS………………


A number is chosen at random from 17 – 30 inclusive. Find the probability that it is a multiple of 5 or an even number.

Wednesday, 9 August 2017

FACTORIZE 2


Factorize 289x- 169y2

Solution

we use difference of two squares a2 – b2 = (a - b)(a + b)

289x2- 169y2  = 172x2 - 132y2

                   = (17x)2 - (13y)2

                   = (17x - 13y)( 17x + 13y)

Hence 289x2 - 169y2 = (17x - 13y)(17x + 13y)

TRY THIS………………………….



factorize 121c2- 49d2

ALGEBRA 3


Multiply 2x - 4y by -5a

solution

= -5a(2x - 4y)

= (-5a x 2x) - (-5a x 4y)

= (-10ax) - (-20ay) [since -5a x 4y = -20ay ]

= -10ax + 20ay    

TRY THIS...................


Multiply 40p - 3q by  -4m


EXPONENTIALS 3


If 52w (40w) = 10w+40 ; Find w.

Solution

52w (40w) = 10w+40

(52)w (40w) = 10w+40

(25)w (40w) = 10w+40

(25 x 40)w = 10w+40

(1000)w = 10w+40

(103)w = 10w+40

103w = 10w+40  (Bases are alike, so they cancel out)

3w = w+40

3w - w = 40

2w = 40

2w = 4020
2       2

w = 20

TRY THIS…………………………….


If 32t (4t) = 6t+10 ; Find t.



VARIATIONS 2


x is directly proportional to y. x=12 while y=4. Find y when x is 69.

Solution

x y

x = ky

12 = k x 4

12 = 4k
4      4

k = 3
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

k = 3, y= ? x = 69.

69 = 3 x y
69 = 3y
3      3

y = 23.

TRY THIS…………….

x is directly proportional to y. x=20 while y=5. Find y when x is 68.


Ans. y=17

LOGARITHMS 3


If log 2= 0.3010; find the value of log 400,000 without using tables.

solution

=log200,000

=log(4 x 100,000)

=log4 + log100,000

=log22 + log105

=2log2 + 5log10

=2(0.3010) + (5 x 1)  [since log10=1 and Log3=0.3010]

=(0.6020) +  5

=5.6020

Hence log400,000=5.6020


TRY THIS……………


If log 2= 0.3010; find the value of log 4,000,000 without using tables.


Sunday, 6 August 2017

ALGEBRA 2


If 13w2 = 325; find w

Solution

13w2 = 325

13w2 = 325               [Dividing by 13 both sides]
13          13

w2 = 25

w = 5          Because the square root of 25 is 5.   

Hence w = 5 .          

TRY THIS……………………….


If 14y2 = 5600; find y.

POLYGONS 2


A regular polygon has 27 sides. Find the total interior angles of that polygon.

Solution

n = 27

Total angles = (n - 2)1800

                   = (27 – 2)1800

                   = 25 x 1800

                   = 45000

Total angles = 45000

TRY THIS……………………….


A regular polygon has 38 sides. Find the total interior angles of that polygon.

EXPONENTIALS 2


If 810 = 32a-2; find a

Solution

810 = 32a-2 

(23)10 = (25)a-2 

230 = 25(a-2) 

230 = 25a-10

230 = 25a-10  [Same bases both sides cancels out]

30 = 5a – 10

30 + 10 = 5a

40 = 5a

40 = 5a
5      5

8 = a

Hence a=8

TRY THIS……………………….


If 510 = 625h-2; find h


QUADRATICS 1


What must be added to x+ 20x to make the expression a perfect square?

Solution

a=1, b=20,c=?

b2 = 4ac

(20)2 = 4 x 1 x c

400 = 4c

400 = 4c
4       4

100 = c

Hence number to be added is 100

TRY THIS……………………

What must be added to x+ 10x to make the expression a perfect square?


Saturday, 5 August 2017

FACTORIZE 1


Factorize completely mc + mr - cr - c2.  

Solution

= mc + mr - cr - c2.

= (mc + mr) – (cr - c2).      [Grouping the factors]

= m(c + r) - c(r + c).   [after factoring out]

 = (c + r)(m - c) answer

  Hence mc + mr - cr - c2. =  (c + r)(m - c).   

TRY THIS..................

Factorize completely pw + pr - rw - w2.

UNITS OF LENGTH 1


Convert 53734m into km.

Solution

1km  = 1000m
  ?     = 53734m

After cross-multiplying;

= 1 x 53734
      1000

=   53734
     1000

= 53.734 km

Hence 53734m = 53.734km

TRY THIS..............................


Convert 72682m into km.

VARIATIONS 1


x is directly proportional to y. x=32 while y=4. Find y when x is 736.

Solution

x y

x = ky

32 = k x 4

32 = 4k    [dividing by 4 both sides]
4      4

k = 8
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

k = 8, y= ? x = 736.

x = ky

736 = 8 x y

736 = 8y
8       8

y = 92.

TRY THIS…………….


x is directly proportional to y. x=40 while y=4. Find y when x is 380.


APPROXIMATIONS 1



Estimate the value of 2.1  x  0.034

solution

= 2.1  x  0.034

= 2.0 x 0.03    [ 2.1 ≈ 2.0 to ones and 0.034 ≈ 0.03 to hundredths]


= 0.06 

TRY THIS...................................

Estimate the value of 5.4  x  0.073

Friday, 4 August 2017

LOGARITHMS 2


If log3(10x - 23)=3; find x.

Solution

log3(10x - 23)=3

 (10x - 23)=33       
                                            
 10x - 23=27

10x = 27 + 23

10x =  50

10x505   
10      10

x = 5

Hence x = 5

TRY THIS…………….

If log3(5x + 6)=4; find x. 

FUNCTIONS 1


Find a linear function f(x) with gradient -6 which is such that f(5)=14.

Solution.

m=-6, points = [5,14] and [x, f(x)]

m = y2-y1/x2-x1

-6 = [f(x) – 14]/x-5

f(x)-14=-6(x-5)  [after cross multiplying]
f(x)-14=-6x+30
f(x)=-6x+30+14
f(x)=-6x+44

A  linear function is f(x) = -6x + 44.

TRY THIS……….

Give out a linear function f(x) with gradient -4 and f(5)=11.


LOGARITHMS 1


Simplify Log2256 - Log3243

Solution

= Log2256 - Log3243

= Log228 - Log335    [Since 256=28 and 243=35].

= 8Log22 - 5Log33    [since Logaa = 1]

= (8 x 1) - (5 x 1)

= 8 - 5

= 3

Hence Log2256 - Log3243 = 3

TRY THIS..................


Simplify Log21024 – Log5625

ALGEBRA 1


If 4x + 5y -9 = 0; find x-intercept.

Solution

x-intercept is when y=0.

4x + 5y -9 = 0

4x + 5(0) - 9 = 0

4x - 9 = 0

4x = 0+ 9

4x = 9
4      4

x= 9/4

Hence x= 9/4

TRY THIS………..


If 11x - 5y - 19 = 0; find x-intercept.


Thursday, 3 August 2017

MATRIX 3



POLYGONS 1


An interior angle of a regular polygon is 780 greater than an exterior angle. Find the interior angle.

Solution

Let i = interior angle, e = exterior angle.

Now i  + e=1800…………………(1)

But i = e+780 …………………(2)

Substitute (2) in (1) above.

e+780   + e=1800

e+ e+780   =1800

2e+ 780   =1800

2e=1800 - 780   

2e=1020

2e=1020          dividing by 2 both sides.
2      2

e = 510

From (1),

i  + e=1800

i  + 51=1800

i  =1800 - 510

i  =1290 answer 

TRY THIS………………………   


An interior angle of a regular polygon is 860 greater than an exterior angle. Find the interior angle.

VECTORS 1





SETS 1


If n(A)= 78 , n(B)= 90 and n(AuB)= 130, find n(AnB).

Solution

n(AuB) = n(A) + n(B) - n(AnB)

130 = 78 + 90 - n(AnB)

130 = 168 - n(AnB)

130 - 168 = - n(AnB)

-38 = -n(AnB)

n(AnB) = 38 [after dividing by -1 both sides]

Hence n(AnB) = 38 answer


TRY THIS………….



If n(A)= 71 , n(B)= 88 and n(AuB)= 130, find n(AnB).

MATRIX 2


EXPONENTIALS 1


Simplify 60
              125-2/3

Solution

60
   125-2/3

= 60 x 1
         125-2/3

=  60 x 1252/3        [Since 1/a-n = an]

=  60 x (1251/3)2        [Since 1251/3 = cube root of 125 = 5]

=  60 x (5)2        

=  60 x 25       since (5)2 = 25

= 1500 answer      
          

TRY THIS…..  

Simplify 20

             216-2/3

MATRIX 1



SIMPLE INTEREST 1


Rayan deposited the amount of 30,000/= in a bank for 4 years and got a profit of 8400/=. Find the interest rate.

Solution

I = 8400/=, P = 30,000/=, T = 4 years, R = ?

I  = PRT
      100

8400  = 30,000 x R x 4
                  100

8400  = 30,000 x R x 4
                   100

8400  = 300 x R x 4
                      
8400  = 1200R
           
78400  = 1200R
1200       1200


Hence the interest rate was  7%

TRY THIS………….


Rayan deposited the amount of 7,000/= in a bank for 3years and got a profit of 1260/=. Find the interest rate?

LOGARITHMS 1




DIFF OF 2 SQUARES 1


Evaluate 18822 – 11182

Solution

We apply difference of two squares: a2 - b2 = (a-b)(a+b).

18822 – 11182= (1882 + 1118)( 1882 - 1118)

                      = (3000)( 628)

                      = 1884000

Hence 18822 – 11182= 1884000


TRY THIS……………………


Evaluate 15442 – 14562